Abstract:We prove that the continuous function${\rm{\hat \Omega }}:2^\omega \to $ that is defined via$X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left( X \right)\,{\rm{d}}X$ is a left-c.e. $wtt$-complete real having effective Hausdorff dimension ${1 / 2}$.We further investigate the algori… Show more
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